Real-rooted atlas

A199478 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(15\,x - 20\,x^{2} + 32\,x^{3} + 16\,n x + 21\,n x^{2} + 8\,n x^{3} - 2\,n^{2} + 3\,n^{2} x + 8\,n^{2} x^{2} - 2\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(30\,x - 25\,x^{2} + 44\,x^{3} + 32\,x^{4} - 2\,n + 39\,n x + 29\,n x^{2} + 28\,n x^{3} + 32\,n x^{4} - 4\,n^{2} + 3\,n^{2} x - 4\,n^{2} x^{2} + 32\,n^{2} x^{3} - 2\,n^{3} - 6\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 15\,x^{2} - 13\,x^{3} - 48\,x^{4} - 64\,x^{5} + 4\,n x + 33\,n x^{2} - 64\,n x^{4} + 2\,n^{2} x + 14\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4Pn(1)
0213
1437
269217
3823839
4104530489
512759220199
614113218888441
71615943231248967

All 15 rows

Technical details
Node id
n-fc3994bf827a75e6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e6576735641c464b

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  2,   1;
  1:  4,   3;
  2:  6,   9,    2;
  3:  8,  23,    8;
  4: 10,  45,   30,    4;
  5: 12,  75,   92,   20;
  6: 14, 113,  218,   88,   8;
  7: 16, 159,  432,  312,  48;
  8: 18, 213,  758,  860, 240,  16;
  9: 20, 275, 1220, 1964, 960, 112;
%F Formula
The row polynomials satisfy -(2*n^3+2*n^2-(2*n^3+3*n^2+16*n+15)*x-(8*n^2+21*n-20)*x^2-(8*n+32)*x^3)*P(n,x) = -(2*n^3+4*n^2+2*n+(6*n^3-3*n^2-39*n-30)*x-(8*n^3-4*n^2+29*n-25)*x^2-(32*n^2+28*n+44)*x^3-(32*n+32)*x^4)*P(n-1,x) + ((2*n^2+4*n+2)*x+(14*n^2+33*n+15)*x^2-(16*n^2+13)*x^3-(64*n+48)*x^4-64*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + x.
%t Mathematica
P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 4*n^2 + 2*n + (6*n^3 - 3*n^2 - 39*n - 30)*x - (8*n^3 - 4*n^2 + 29*n - 25)*x^2 - (32*n^2 + 28*n + 44)*x^3 - (32*n + 32)*x^4)*P[n - 1, x] + ((2*n^2 + 4*n + 2)*x + (14*n^2 + 33*n + 15)*x^2 - (16*n^2 + 13)*x^3 - (64*n + 48)*x^4 - 64*x^5)*D[P[n - 1, x], x])/(-2*n^3 - 2*n^2 + (2*n^3 + 3*n^2 + 16*n + 15)*x + (8*n^2 + 21*n - 20)*x^2 + (8*n + 32)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 4*n^2 + 2*n + (6*n^3 - 3*n^2 - 39*n - 30)*x - (8*n^3 - 4*n^2 + 29*n - 25)*x^2 - (32*n^2 + 28*n + 44)*x^3 - (32*n + 32)*x^4)*v[i - 1] + ((2*n^2 + 4*n + 2)*x + (14*n^2 + 33*n + 15)*x^2 - (16*n^2 + 13)*x^3 - (64*n + 48)*x^4 - 64*x^5)*deriv(v[i - 1], x))/(-2*n^3 - 2*n^2 + (2*n^3 + 3*n^2 + 16*n + 15)*x + (8*n^2 + 21*n - 20)*x^2 + (8*n + 32)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.